Regularity Criteria And Global Well-posedness Of Fluid Dynamical Equations

This paper is divided into two parts.In part one we consider the regularity criteria and global well-posedness of two types of hydrodynamic equations,the incompressible Navier-Stokes equations(NSE for short)and incompressible magnetohydrodynamic equations with damping(MHD with damping for short).In part two we generalize Euler-Rodrigues rotation formula of three Euclidean space to manifold with metric which dimension is more than 2 and the sign difference is constant.The details are as follows.In the first chapter,we briefly introduces the research background,the progress,the motivation,the contents,the methods and innovations of NSE and MHD with damping and Euler-Rodrigues rotation formula.In the second chapter,we study the regularity criterion of NSE based on the ratio improvement of Prodi-Serrin-Ladyzhenskaya(PSL for short)type.We first establish two new ratio regularity criteria,in which the physical quantities on molecules belong to Serrin type and the velocities in denominators are Serrin type critical norms through some new estimation methods and techniques.Finally,for 0<?<1,we obtain the regularity of NSE on ?/(|u|?)(or equivalently,on ?_u/(|u|?)).In the third chapter,we study the criterion of component regularity of NSE.For this reason,we first establish two new new multiplicative Sobolev inequalities,and then prove three new criteria of component regularity with the help of these two inequalities.In the fourth chapter,we focus on the more complex theory of regularity and uniqueness of solutions of MHD with damping.By virtue of sophisticated inequality techniques,this chapter proves that MHD with damping exists a unique global strong solution if one of the four conditions holds.In the fifth chapter,we extend the Euler-Rodrigues rotation formula of three Euclidean space(R³,?_ab)to manifold with metric(M,g_ab)which dimension is more than 1 and the sign difference is constant.As an application,in Minkowski space(R⁴_(1,3),?_ab)we investigate Lorentz transformation of elastic collision of relativistic particles.In the sixth chapter,we summarize the work of this paper and look forward to the future research.